Root coordinate cells and generation

Let $H$ be the subgroup generated by $T$ and the $G_{\alpha }$ for $\alpha \in \mathrm{\Delta }$. Each $G_{\alpha }$ contains its two root groups and a representative of $s_{\alpha }$, by the rank-one results in [F1]. As the simple reflections generate $W$, products of these representatives in $H$ represent every $w\in W$. Conjugation by them and $W\cdot \mathrm{\Delta }=\mathrm{\Phi }$ show that $H$ contains every root group. It therefore equals $G$ by [F1]. Each $G_{\alpha }$ is generated by $T$ and its two root groups, so this also proves generation by $T$ and the simple positive and negative root groups. If $G$ is semisimple, let $K$ be generated by all root groups. The standard ${\mathrm{S}\mathrm{L}}_2$ root-group generation in each rank-one derived subgroup puts every coroot image in $K$. The coroots span $X_{\ast }(T)\otimes \mathbb{Q}$ for semisimple $G$, so these images generate $T$. Thus $K=G$. If $\mathrm{\Delta }$ is empty, $G=T$ and the first assertion still holds; a semisimple group with empty root system is trivial.

The subgroups $U_w$ and $U^w$ are smooth connected $T$-stable by [F2], being generated by root groups with weight sets ${\mathrm{\Phi }}^{+}\cap w{\mathrm{\Phi }}^{+}$ and ${\mathrm{\Phi }}^{+}\cap w{\mathrm{\Phi }}^{-}$ respectively; these two subsets of ${\mathrm{\Phi }}^{+}$ are disjoint and their union is ${\mathrm{\Phi }}^{+}$ (a positive root is either in $w{\mathrm{\Phi }}^{+}$ or in $w{\mathrm{\Phi }}^{-}$), and each is ordinarily closed: a positive root sum of members remains positive and its inverse image under $w$ retains the same positive or negative sign. Ordinary closure is sufficient by [F2], so each is the weight set of a smooth $T$-stable subgroup of $U$ and the product decomposition of $U$ from [F2] gives a $T$-equivariant isomorphism $U_w\times U^w\to U$. The membership criterion for root groups follows because $U_{\alpha }\subseteq U_w$ iff the weight $\alpha$ belongs to the weight set of $U_w$ (uniqueness of the smooth subgroup with a given weight semigroup in [F2]). This proves (b).

The stabilizer of $n_{w}B$ in $U$ is $K=U\cap n_{w}B{n}_w^{-1}$. Work geometrically and take $K_{\mathrm{r}\mathrm{e}\mathrm{d}}^{\circ }$, which is smooth connected over the algebraically closed field. By the root-subgroup containment criterion its root groups are exactly those with $\alpha \in {\mathrm{\Phi }}^{+}\cap w{\mathrm{\Phi }}^{+}$, since its Lie algebra is contained in the Lie intersection of [F4], and all these shared root groups lie in $K$. The ordered-coordinate theorem therefore identifies $K_{\mathrm{r}\mathrm{e}\mathrm{d}}^{\circ }$ with $U_w$. Its dimension is the dimension of $\mathrm{Lie}K$, so $K$ is regular at the identity and translation makes it smooth. To identify all components, choose a regular cocharacter $\lambda$ with $n_{w}B{n}_w^{-1}=P_G(\lambda )$. Every root coordinate of $U$ has nonzero $\lambda$-weight. An element of $U\cap P_G(\lambda )$ has a limit under conjugation, so its negative coordinates vanish and the positive coordinates tend to zero. Thus its limit is $1$, and it lies in $U_G(\lambda )$. This also contracts $K$ to $1$, showing its components all meet the identity component, hence $K$ is connected. Therefore $K=U_w$, and the equality descends to $k$. Order the complementary root groups first: $U^w\times U_w\simeq U$ identifies the fppf quotient $U\mathrm{/}{U}_w$ with $U^w$, since right multiplication by $U_w$ changes only the second factor. The homogeneous orbit map identifies this quotient with $U{n}_{w}B\mathrm{/}B$, and its dimension is $\dim U^w=n(w)$.